Math 360, Fall 2019, Assignment 1
From cartan.math.umb.edu
By one of those caprices of the mind, which we are perhaps most subject to in early youth, I at once gave up my former occupations; set down natural history and all its progeny as a deformed and abortive creation; and entertained the greatest disdain for a would-be science, which could never even step within the threshold of real knowledge. In this mood of mind I betook myself to the mathematics, and the branches of study appertaining to that science, as being built upon secure foundations, and so, worthy of my consideration.
- - Mary Shelley, Frankenstein
Read:[edit]
- Section 0.
Carefully define the following terms, then give one example and one non-example of each:[edit]
- Binary operation (on some set $S$). (Note: so far we have treated the subject only at the level of an overview, so you will need to be a little vague. The same is true of the next three definitions, as well as problems 2-5 below. Do the best you can. We will give proper definitions as the semester proceeds.)
- Identity element (for a given binary operation).
- Inverse (of an element, relative to a given operation).
- Symmetry (of a geometric figure).
- Subset (of a given set).
- Improper subset (note that we did not discuss this in class, so you may need to look in the book for the definition).
- Empty set.
- Cartesian product (of two sets).
Carefully state the following theorems (you do not need to prove them):[edit]
- Russel's paradox (this is not really a theorem, but it is an important fact).
- Axiom of extensionality (relating equality of sets to mutual containment; again, not really a theorem).
- Counting principle (relating the size of $A\times B$ to the sizes of $A$ and $B$).
Solve the following problems:[edit]
- Section 0, problems 1, 2, 3, 4, 5, 6, 7, 8, and 11.
- How many symmetries does a square have? Describe them.
- Working in the "symmetry group" of a square, how many solutions does the equation $x^2=e$ have? What about the equation $x^4=e$?
- How many symmetries does a regular $n$-gon have? (This means an $n$-sided polygon, all of whose sides are equal and all of whose angles are equal.)
- Working in the symmetry group of the regular $n$-gon, how many solutions does the equation $x^2=e$ have? What about the equation $x^n=e$? (Warning: the answers depend on whether $n$ is even or odd.)
